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Results tagged with quadratic-forms
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user 38624
Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.
4
votes
Can a positive binary quadratic form represent 14 consecutive numbers?
Here's a heuristic that suggests why arbitrarily large strings of consecutive
numbers should be representable by some binary quadratic form. For simplicity
consider a prime $p$ that is $3\pmod 4$ s …
- 43.2k
7
votes
Accepted
Achieving consecutive integers as norms from a quadratic field
The answer to Speyer's question as stated is no, this need not be
the case. To see this let $p\equiv 3\pmod 4$ be a prime and consider
the associated imaginary quadratic field ${\Bbb Q}(\sqrt{-p})$ …
- 43.2k
8
votes
Counting fundamental units of real quadratic fields
In Proposition 4.1 of Sarnak you'll find an asymptotic for the related quantity (take there $p=1$) when one considers all ring discriminants (not just the fundamental field discriminants that you want …
- 43.2k
6
votes
Questions on $x^2+y^2+z^2$, $x^2+y^2+2z^2$ and $x^2+2y^2+3z^2$
Problems like Question 1 go back to Euler and Gauss. This question is essentially asking which idoneal numbers are sums of two squares. There is a famous open problem going back to Euler and Gauss a …
- 43.2k
19
votes
Accepted
Upper bound on answer for Pell equation
Let $d$ be a positive fundamental discriminant, $\epsilon_d$ denote the fundamental unit, $h(d)$ the class number, and $\chi_d$ the primitive character associated to the discriminant $d$. The class n …
- 43.2k
19
votes
Accepted
Many representations as a sum of three squares
The formula for $r_3(n)$ essentially connects this with a class number of an imaginary quadratic field, or (apart from the $\sqrt{n}$ scaling) with the value of an $L$-function at $1$. So your questi …
- 43.2k